On the Group Ring of a Finite Abelian Group

BULL. AUSTRAL. MATH. SOC. MOS 2080

VOL. I (1969), 245-261.

On the group ring of a finite abelian group

Raymond G. Ayoub and Christine Ayoub

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given - without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G . It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

Introduction

In this paper we study the group ring of a finite abelian group G over the field Q of rational numbers and over the (rational) integers Z . We give a new proof of the well-known fact that the group ring QG is a direct sum of cyclotomic fields [G. Higman] which enables us to say which fields arise and with what multiplicity (see also [5]). We show that the projections of QG onto these fields are determined by the inequivalent characters of G . This proof is to our mind conceptually much simpler than the one using representation theory. Granted an acquaintance with the elementary properties of tensor products (of commutative algebras) all that is needed is developed in the paper itself.

In §3 we consider the group U of units of ZG ; we find that U is the direct product of a finite group and a free abelian group F of finite rank - a fact already established by Higman (cf. [4]) - but our results

Received 21 March 1969. Received by J.. Austral. Math. Soc. 17 September 1968. Revised 17 December 1968. Communicated by G.B. Preston. This research was done under NSF Contract GP-5593. 245

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enable us to calculate the rank of the group F (Theorem h). Theorem 5 gives a formula for the orthogonal idempotents of QG . Finally in the last paragraph we study the embedding of Zff in the direct sum of cyclotomic fields, in the case when G is cyclic of prime power order.

Definitions and notations

We list the notations which will be used throughout the paper. Some of these are standard, while others are our own invention - most of the latter will be explained in the body of the paper but are listed here for the convenience of the reader.

® is used for the direct sum of rings.

If X. are rings and if x. € X- (for i e J), we write $ x- for

the element of © X. whose i-th component is x. . iel % T-

8 denotes the tensor product over the rational field.

If i? is a ring, we write mR for Rffi . . . ® R (m terms) and R*" for R 8 ... 8 if (m terms).

Q is the field of rational numbers.

?, denotes a primitive (complex) d-th root of unity. QjCx) denotes

the cyclotomic polynomial satisfied by the primitive d-th roots of unity over the rationals.

Q, is the splitting field of $Jx) , so that 0, is obtained from

Q by adjunction of £, .

F[x] denotes the ring of the polynomials in x over the field F and F(a) the field obtained by adjunction of a to F .

If fix) e F[x] , deg (f(x)) = degree of fix) , (fix)) = ideal generated by f(x) .

The exponent of a finite group G is the least positive integer m

such that gm = 1 for all g 6 G . (a) is the cyclic group generated by a , and o(a) = order of a .

x denotes the direct product of multiplicative groups.

We write our mappings on the left so that a o 3 denotes the mapping 3 followed by the mapping a .

For the homomorphism n , Kerfnj = kernel of n and (r)\G) = restriction of n to G {G a group).

N and Z denote the natural numbers and integers, respectively.

Z, denotes the polynomials in £,, over Z .

Modz(S} = module generated by S over Z .

QG • and ZG are the group rings of G over Q and Z , respectively.

Actually only finite abelian groups are considered - but this is explicitly stated in the theorems.

1. The structure of QG

DEFINITION 1. If c, is a (complex) primitive d-th root of unity,

QJ = Q^rf) • We note that Q-, does not depend on the particular d-th

root of unity chosen.

PROPOSITION 1. Let G be a cyclic group of order n , and Q the field of rational numbers. Then

(i) QG =

Proof. Let $-,(x) be the cyclotomic polynomial satisfied by the

primitive d-th roots of unity over Q . Then

QG = Q[x]

= ® Q[x] d\n

But Q[x] / ' = Q($J = Qj . Hence (1) QG = © Q, . d d d /(*d(x)) d\n

Note. If G is generated by a , then under the isomorphism (l),

(2) a* ++ ® £t 9 QA . d\n d d\n d

PROPOSITION 2. If the groups A. (1 < i < k) are subgroups of the

group G such that G = A i x ... x A^ (direct product) then

(3) QG = QAi 8 ... 8 QA, (tensor product over Q) .

Note. If Z denotes the integers then we also have

ZG - ZAX 8 ••• 8 ZA, . The proof follows in the same way.

Proof. If g = a\ . . . a, , where a. £ A. (1 < i < k) , map g onto

a\ % ... 8 a-, e QA\ 8 ... 8 QA, . This mapping sends a basis for QG onto

a basis for QA\ 8 ... 8 QA, and hence we can extend it to an isomorphism.

PROPOSITION 3. Q, 8 Qy - direct sum of <$>(d) copies of Qm = $(d) Qm

where d = g.c.d. (k,l) , m = l.c.m. (k,l) and <(> denotes the Euler §-function.

s t Proof. Let m = ks = It , where (s.t) = 1 . Then C, and C are mm clearly primitive k-th and l-th roots of unity respectively. Thus the field Q(Z>-K3^-i) , obtained by adjoining to Q primitive fe-th and Z-th

roots of unity, is contained in Q ; on the other hand, if we choose u

U V anind thev fielsucdh thaQ(£,^C,y)t us + vt= Qj,(^i^= 1 , the • nTherefore Z^ = (fy, w(fye hav e showand nhenc thaet C,m is

Now let

(5) 9z(x) = fi(x)...fs(x)

be the decomposition of the cyclotomic polynomial into irreducible factors

in Qv[x] • Since by (h) Q is obtained from Q, by adjoining any root K. 171 K. of 7 (x) (and therefore, any root of any f.(x))

Q - Qhix] / (1 < i < s) . Thus each f.(x) has degree m k /(f^x)) = =

[Qm:Qk} = ^j-Jjgj.-lffi,

= $(m) <\>(d) . Also

(l) = deg (

so that (6) s = cfxu> .

From a known theorem we have that Q, ® Sv = 6^[^] / > an(i

from the decomposition (5) and (6) this is isomorphic to

Q , since Q-Ax] / - Q for

1 < i < §(d) .

Note. Applying Proposition 3 several times we could establish that

for positive integers k\3...,k

Q, 8 ... 8 Sir, = s Q J

where m = l.c.m. fti,,..^ J and s is a positive integer (which we could

calculate). In particular^ if g.c .d. (k -,k •) = 1 for i 4 3 , i 3 (7) Q, % . . . % Q, = Q (i.e. s = i).

THEOREM 1. If G is a finite abelian group of exponent m , there exist integers u •, > 0 such that

(8) QG = ® w, e, .

Proof. Let G = i4i x ... x A, , where A • is cyclic of order d • for

1 < i < k and d\ \ di \ . .. | d-, = m . Then by Proposition 2,

§G = QA\ 8 ... 8 Q4, and hence by Proposition 1, K QG - 9 Q •,%... 8 9 Q •,• Now if we expand using the distributive law

d\di d\dk and the note after Proposition 3, we see that QG is the direct sum of

cyclotomic fields Q, , where each d divides m .

2. The projections of QG DEFINITION 2. Let G be a finite abelian group. The characters x and \p are equivalent if Kerfy/1 = Ker (\p) . LEMMA 1. Let G be a finite abelian group of order n and exponent m ; let d divide m and let t-, be the number of cyclic subgroups of G of order d . Then

(1) The number of inequivalent characters x such that x(G) = (t, J

is td .

(2) If x(G) = (Z,-,) j the number of characters equivalent to x ^s

(3) I t d\m u

Proof. (1) Let D be a subgroup of G such that G/D is cyclic of order d , and let C, be a primitive d-th root of unity. Then there

exists an epimorphism from G to (^j) with kernel D - i.e. a character

with kernel D . Therefore, the number of inequivalent characters x such that x(&) = (ZJ) is equal to the number of subgroups D such that G/D

is cyclic of order d and this is equal to the number of cyclic subgroups of order d .

(2) Let X be a character with x(G) = (Zj) and D = Ker(x) • Then

for a in the automorphism group of (ZJ) ty - a ° X is a character of

G with kernel D - i.e. ^ is equivalent to X ~ and distinct

automorphisms a give rise to distinct characters \p . Furthermore, any

character ty of G which is equivalent to X is of this form, since the mapping a defined by a(x(g)) = ty(g) for g e G is an automorphism of (QJ) • But the automorphism group of (CJ-' has order (d) . Thus there

are §(d) characters equivalent to x •

(2) From (2) and (3) the total number of characters of G is

tj <$>(d) . Hence n = T t •, §(d) , since G has n distinct d\m d characters.

COROLLARY. If XIJ—>X-i are inequivalent oharaaters of the finite I abelian group G of order n and if \ §(d •) = n } where \AG) = (£J ) i=l i for 1 < i < I j then the characters \-(l < i < I) form a complete set of inequivalent characters of G . THEOREM 2. Let G be a finite abelian group of exponent m . Then

QG = © tjQji where tA is the number of cyclic subgroups of G of d\m d d d order d . The projection IT of QG onto the component Q, defines a character x = (v\&) °f G onto ft; J and the characters so defined form a complete set of inequivalent characters of G . Proof. From Theorem 1, we have (8) QG - 9 U-, Q, . d\m d d It is clear that for each projection ir onto a component Q, we get a character \ = (n\G) of G . We show first that if TTj and T^ are projections onto different components then the characters x = ^1 1^ an^-

\j) = (Tr2|

T,d into ad;d) .

Since QG - 9 t-,QA, there is an element r e QG with Tt\ (r) = 0 d\m d d

and ^x(r) % 0 . Now r = \ r(g)g with rig) e Q and we have: geG 0 = IT! (r) = I r(G) TTi(g) = £ r(g) \(g) . On the other hand, geG geG

0 =(= i\2(r) = I rig) v2(g) = I rig) \p(g) = J r(g) (a o gzG geG geG

= af I r(g) X(gA = 0 .

This is clearly impossible so that x and IJJ cannot be equivalent.

Let irlif. . .,TU be the projections of QG defined by (8), and let

Y- = (ir-\G) for 1 < i < I . We establish the fact that the v- form a

complete set of inequivalent characters of G . Clearly if x • (O ~ (ZJ ) » \ l TT . (QG) = Q, since G generates QG over Q . Hence QG = 9 Q, so 1 ai i=l di I that comparing dimensions we have: n = T §(

to Lemma 1, the v- are a complete set of inequivalent characters of G .

Thus also

(9) QG ~- d.a td Qd .

REMARK. It may happen that Q, = Q,, with d < d' - in fact, this is

the case if, and only if, d is odd and d' = 2d . It can be shown that the expression (8) (obtained by use of Propositions 1-3) is the same as (9) - i.e. Uj=t_,. However, the proof is a little tedious and so we thought

it not of sufficient interest to include.

3. The units of ZG

THEOREM 3. (G. Higman) The only units of finite order in ZG are of the form ± g (g € G) .

Proof. Let u e ZG be a unit of finite order. Then if 7T is the projection of QG in the isomorphism QG - 9 t-,Q-,, T\(u) is a unit of d\m d a finite order in Q-, - i.e. irfuj is a root of unity. Thus if

u = I u(g)g with u(g) e z , T\(U) = I u(g) v(g) = \ u(g) \(g) = p , ge.G geG geG where x = (it\G) and p is a root of unity. Since these characters X X form a complete set of inequivalent characters we have:

f (10) I u(g) x(g> = Px or every

the character group G and where each p is a root of unity. Using the X orthogonality relations of the characters, we obtain nu(g) = \^ P

for g e G and from this it is easy to deduce that there exists a g £ G

such that u(g ) = ± 1 and u(g) = 0 for g $ g (cf. e.g. [/]). Thus

u = ± g .

THEOREM 4. Let G be a finite abelian group of order n and let U denote the group of units of ZG . Then U = T * F with T = {± g \ g e G)

and F free abelian of rank -J-fn + 1 + t2 - 21) , where ti = number of elements of G of order 2 and I = nvmber of cyclic subgroups of G .

Proof. ZG is isomorphically embedded in 9 t, Z, under the d\m d d isomorphism (9) (here Z, = Z(Z,,)). Since "both ZG and 9 t, Z, are d d d\m d d free abelian of rank n , ZG is of finite index A: in 8 4, Z, d\m d d (identifying ZG with its image under the isomorphism). Let U denote the group of units of ZG and f/i the group of units of ® t , Z, . Then U\

is generated by the units of the components Z, - and the unit group of each

Z, is finitely generated by Dirichlet's Unit Theorem (cf. [3] p. 12U Satz

100; we note that Z, is the ring of integers of 6, - cf. [6], p. 2&h,

7-5-U Theorem). Now we show that U\/U is finite. It is sufficient to show that every unit u of Z, has finite order, mod Z(G) . Letting

A = principal ideal generated by k in Z , , we have u — = 1 (mod A)

where $(£) denotes the number of reduced residue classes, mod A • Hence

( } J(h) = j + xk (cf. [3], Satz 81+), where X e Z^ and so J = e ZG since

[9 t, Z, : ZG] = k . Hence U = T * F , where T is finite and F is

free abelian of rank = the torsion-free rank of U\ . In Theorem 2, we have determined T so it remains to calculate the torsion-free rank of U\ .

By Dirichlet's Theorem if d > 2 , Z, has "' - 1 independent units

of infinite order- Thus the torsion-free rank of JJ\ is , /J. -I (*^Y~ - 1 • But Y t, (d) = n , and \ t,= I = number of *.,... . ' d\m

cyclic subgroups of G and t2 = number of elements of order 2 (by Theorem

l). An easy calculation shows that

dim

4. The idempotents of QG

In what follows we will identify QG with © t, 0, (using the

isomorphism of Theorem 2). If 0 is a subgroup of G with G/Z3 cyclic,

then irD will denote the projection of QG onto 0, whose associated

character Xn has kernel D . By Theorem 2, there is exactly one such

projection. Thus if u = ® s_ , then s_ = ^D(u) . It is clear that w is

idempotent if, and only if, each S- is either 0 or 1 - i.e. each

idempotent is a sum of primitive idempotents w_ , where TT_fM_j = 1 and

ir_, (M-,) = 0 for D' ^ D . In Theorem 5 we obtain the representation of

the idempotents w as elements of QG .

THEOREM 5. For D

primitive idempotent which corresponds to the identity of ir^fQG) , and for

K < G let ey = \ un . Then K

(11) e = 4r I k K 1*1 keK

(12) u_ = I V (\B/D\) e , U D

where u is the fSbius ^-function.

Proof. Consider the element rv = -nrr I k e QG . For H < G with K 1*1 kcX

G/H cyclic,

•2 if K < H ,

otherwise

•1 if K < H , IT (U ) = f K

Hence ev = rv = -rrrr 1 k , as claimed. K K I*' keK

(12) follows from the Mobius inversion formula. -1

COROLLARY. Let G = (a) be oyolia of order pn (p a prime), and let

1 1 1 = -±z7 P "I- \a?( if J e QG, 0 < i < n , (13) € z I

Then

9 2- ,7=0 p3

J Z . .

5. The index of Z(G) in • td Zd

THEOREM 6. Let G = (a) be a ay alia of order pn (p a prime) and

let a. be defined by (13). Then the set {e- a3)

i n (0 < i < n , 0 < j < $(p )) is a basis for ffi Z . over Z . (Here we 3=o p3

1 We are indebted to the referee for the use of the Mobius u-function

in the representation of un and for pointing out that the Mobius inversion formula holds under these circumstances.

7 n are identifying e -ar with its image in ffi Z .) .

3=o p3

Proof. Let M = mod_ {e • cP). . -By the Corollary to Theorem n . n 5, M < ® Z . since e. and a*7 £ © Z . . The number of generators 3=0 p3 V 3=0 p3 n . n e^ a? is I §(p3) = pn . Thus if we establish that M = © Z . , the J=o 3=0 p3 theorem will be proved.

We prove by induction on i , that e. ar and u- cr (where % If 6 u. = e • - e . -J are in M a V j N . Note that % t> if—i

(15) a"7' = 1 © p^ » ... © pf. © ... © a0'

where p. £ Z . is a primitive p -th root of unity (strictly speaking the

"=" should be "«-»•" but we are identifying ZG with its image). If

3 e 3 i = O,eaP=u a = e £ M 3 V j il? . So we assume that e • ^ a and

3 3 u- n a? €. M, M j e N . u. a = 0 ® . .. ® p . ©...90 and Z(p J is

generated over Z by 1, p., p?, ..., p? P ; hence u • a3 €. M , \/ j e N

if u. a3 & M for 0 < J < (JiCp1"^ . So consider u. a3 with

0 < 3 < ^(p1) • We have

3 J 7 (16) u- a ' = e. a ' - e. 7 a" '

and e • a £ M since it is one of the generators of M , e • , a3 £. M by

the induction assumption. Hence u- tf £ A/ . Also e- = u. + e • 7 and hence

(17) e. a3' = u. J -h e. , a3' e M, V j eff .

THEOREM 7. Let G = (a) be cyclic of order pn (p a prime) and let &1 (0 < i < n) be defined by (13). Then

(A) e- a3' has order pn~v , mod ZG CV j e N) .

(B) The elements e • a3 CO < i < n-1 , 0 < j < ^(pV)) form a basis 1, — — — n for ® Z . , mod ZG . j=o p3

n (C) 9 Z . : ZG 3=0 p3

Proof. By the previous theorem, the e.or (0 < i < n 3 0 < j < <$>(p )) n form a basis for ® Z . = M . Since e a3 = a3 e ZG , the elements j=o p3 e. a3 (0 < i < n-1 , 0 < j < $(p1)) generate M , mod ZG . It is clear

from the definition of e• , that e. a3 has order pn~% (V'3 e N) .

Hence (A) holds. To prove (B), we need only show that the

e • ar (0 < i < n-1 , 0 < j < fp )) are linearly independent, mod ZG . "2, — — — We prove this by induction on n . If n = 1 , there is just one generator e and so there is nothing to prove.

How assume the result true in case of a cyclic group of order p

Let B = (b) , where b = aP ; B < G has order pn~ and so if we let

k i * v -r/ W . = —-, ;—r ) p k=o

the induction hypothesis implies that w. b3 (0 < i < n-2 , 0 < j <

are linearly independent, mod ZB . But

j 2 +1 k Wi = n_ (i+v* i ~ \y ) = ei+1

J 1 and w. iP = e. ? aP . Hence the elements e. aP' (0 < i < w-2 ,

0 < j < (jifp )) are linearly independent, mod ZB . Now assume

n ($(p )-

Multiplying by p we get:

n 1 n n (19) P oQo eQ + p [U aio. p"- e.} = p t -p C

n (p -l CQ0 I I eflep ZG (using (l8): "• i=o c = 0 (mod p) .

So we can set a = p a' with a' e Z . Noticing that

-r \ a = ej + e\ a + ... + e\ aP~ , we can write (19) in the form

1 (20) a'QQ (ex + ... + ei aP' ) + I I o^. e' cP = t e ZG . i=l 3=0

Consider now the terms involving only the cP with j = p - 1 (mod p) . We get

n-i

jlp-iCmod p;

is—a L=O

= t1 a^"2 with t' e ZB .

But as we saw above the e. , a "

i i. i-1 (0 < i < n-1 , 0 < I <

Therefore a' = 0 (mod p ) and

P~l) for 2 < i < n-1 , C

Since c = p e1 , c =0 (mod pn) and we obtain oo e oo oo r

T T a.. el J & ZG . L • n ._ i-3

Repeating the argument (with p-1 replaced by s) we obtain

n V (21) a. , = 0 (mod p ~ ) for: 1 < i =< n-1 i3 up+s = 0 < I < §(pL~ ) 0 < s < p-1 .

But if 0 < j

0 < s < p-1 , and j = tp + s with 0 < 1 < ^(pV ) . Hence from (21) we deduce that

e. . = 0 (mod pn~V) , 1 < i £ n-1 , 0 < j < ^(p1)

and this is what we wanted to show. Thus (B) is established.

Finally we consider (C). Since e. cP has order pn V , mod ZG, and n these elements form a basis for ffl Z . = M , mod ZG , the order of the 3=0 p° quotient group M/ZG is

PH (Pn~

But since n + (n-1) §(p) + ... + $(p ) = 1 + p + ... + pn (this can easily be established by induction), we have r:

COROLLARY. Let G = Ca; be cyclic o/ order pn . 1/ we define

n-i , . k p -1 r l\ t. = £ Ml 0 < t ~ ~ (0 < i < n , 0 < 3 < i/(pV)) form a basis for ZG (over Z) .

Proof. Let

Ml =Modz ri - J0 < t < „

0 1 3 < ~~(p") •~~

~~Clearly My < ZG < M = ffi Z (e^ a?) . Let r e ZG ; then 0~~

~~r = y o . . e • cP with a . . € Z so that a . . = a'. • p with~~

~~er. . S Z since the e. ar are linearly independent, mod ZG . Thus~~

~~n V r = I a'- • t. cP e M\ , since p ~ e- = t- . Hence Mx = ZG . But the~~

t. a*7 are linearly independent since the e. a*7 are; thus If 1*

~~PROPOSITION 4. Let B. < A. be free abelian groups of rank n. and~~

~~let A-/B. have elementary divisors d.n, ...,d- (1 < i £ r) . Then r A\ 8 .. . 8 A ,„ „ „„ has a basis of II n- elements whose orders r/Bi&vB t~~

~~are d^^-.-.d^ , where 1 < 3i < ^ .~~

~~Proof. Let A- have a basis a-,,...,a- and B. a basis ^~~

~~bil=dilaiV->bin. =din.ain. {for ^ ^ r) • Then ^1 « • • • 8~~

~~has a basis a^ . 8 .... 8 a . (2 < j\\ < n^) anandd B! 8 ... 8 By has a~~

~~basis bldi , ... 9 fc^ = ^^ ... ^.j^^ 8 ... e SiJ H < J. < ,J~~

~~and the Proposition follows.~~

~~COROLLARY. Under the hypothesis of the Proposition,~~

~~[Ax 8 ... 8 A : fix 8 ... 8 B ] = II il~~

~~If G is a finite abelian group, and G = C\ * ... x C, , where each~~

~~C- is cyclic of prime power order, then ZG - ZC\ 8 ... 8 ZC, (note after~~

~~Proposition 2) and for 1 < % < k , ZC is isomorphically imbedded in~~

~~ffi Z j = M. , where Ic.l = p. . Theorem 7 tells us the structure of~~

~~the finite additive (abelian) group Af. / . Hence by using Proposition v /zci~~

~~h we could calculate [M\ 8 ... 8if, ; ZG] . However, this does not give~~

us the index of ZG in 8 t , Z, since Mi 8 ... 8 M, is only imbedded j i u a K. d\m in this last group as a subgroup of finite index. The calculation of this index would seem to be rather complicated. References

~~[I] S.D. Berman, "On the equation x = 1 in an integral group ring", Ukrain. Mat. Z, 7 (1955), 253-261.~~

~~[2] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, (Interscience, New York, 1962).~~

~~[3] Erich Hecke, Vorlesungen uber die Theovie der algebraischen Zahlen, (Chelsea Publ., New York, I9U8).~~

~~[4] Graham Higman, "The units of group rings", Proa. London Math. Soc. (2), 46 (191*O), 231-2U8.~~

~~[5] Sam Perl is and Gordon L. Walker, "Abelian group algebras of finite order", Trans. Amer. Math. Soc. 68 (1950), 1*20-1*26.~~

~~[6] Edwin Weiss, Algebraic number theory, (McGraw-Hill, New York, 1963).~~

~~Pennsylvania State University, University Park, Pennsylvania, USA.~~